Dividing Polynomials; Remainder and Factor Theorems

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Dividing PolynomialsRemainder and Factor
Theorems
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Objectives

• Use long division to divide polynomials.
• Use synthetic division to divide polynomials.
• Evaluate a polynomials using the Remainder
  Theorem.
• Use the Factor Theorem to solve a polynomial
  equation.

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How do you divide a polynomial by another
polynomial?

• Perform long division, as you do with numbers!
  Remember, division is repeated subtraction, so
  each time you have a new term, you must SUBTRACT
  it from the previous term.
• Work from left to right, starting with the
  highest degree term.
• Just as with numbers, there may be a remainder
  left. The divisor may not go into the dividend
  evenly.

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Example

• Divide using long division. State the quotient,
  q(x), and the remainder, r(x).
• (6x³ 17x² 27x 20) (3x 4)

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Example

• Divide using long division. State the quotient,
  q(x), and the remainder, r(x).

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Remainders can be useful!

• The remainder theorem states
• If the polynomial f(x) is divided by (x c),
  then the remainder is f(c).
• If you can quickly divide, this provides a nice
  alternative to evaluating f(c).

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Synthetic Division

• Quick method of dividing polynomials
• Used when the divisor is of the form x c
• Last column is always the remainder

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Example

• Divide using synthetic division.

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Example

• Divide using synthetic division.

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Factor Theorem

• f(x) is a polynomial, therefore f(c) 0 if and
  only if x c is a factor of f(x).
• Or in other words,
• If f(c) 0, then x c is a factor of f(x).
• If x c is a factor of f(x), then f(c) 0.
• If we know a factor, we know a zero!
• If we know a zero, we know a factor!

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Zero of Polynomials

• If f(x) is a polynomial and if c is a number
• such that f(c) 0, then we say that
• c is a zero of f(x).
• The following are equivalent ways of saying the
• same thing.
• c is a zero of f(x)
• x c is a factor of f(x)

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Example

• Use the Remainder Theorem to find the indicated
  function value.